The long range objectives of this research are to further the understanding of the dynamical behavior of aggregates of interacting cells, and to apply this knowledge to problems of pattern formation in developmental biology and to problems in physiology. The proposed research falls into three major categories: (1) studies on signal detection, adaptation and aggregation in Dictyostelium discoideum, (2) studies on synchronization, phase-locking and other phenomena in coupled cells, and (3) studies on pattern formation in generalized Turing systems. The specific aims in (1)\are to extend a model of adaptation in Dictyostelium discoideum to incorporate the dynamical behavior of the cyclic GMP network, to incorporate adaptation and relay in a model for signaling and chemotaxis, and to develop continuum descriptions of aggregation. The objectives in (2) are to understand the dynamics of coupled cells, both forced and unforced, under various modes of coupling, and to analyze a realistic model of pacemaker cells with a view toward understanding simple models of caridac arrhythmias. The aim in (3) is to study reaction-diffusion equations with modulated diffusion coefficients to determine the sensitivity of spatial pattern formation in such models to changes in the size and shape of the developing tissue. The approach to these problems will be as follows. First the relevant experimental literature will be analyzed to provide guidance in the formulation of the mathematical models. Next the governing equations will be analyzed to develop a qualitative understanding of the model and its parametric sensitivity. Finally the necessary analytical and numerical techniques for solving the equations will be developed, and the numerical simulations involved will be done. The work on signal relay and adaptation will lead to a better understanding of signaling, chemotaxis and aggregation in Dictyostelium discoideum, of chemotaxis in other systems, and of the dynamics of calcium-cyclic nucleotide networks. The studies on pattern formation will contribute to the understanding of both normal and abnormal development in biological systems, and in particular, to the understanding of how feedback control of interactions between cells affects the stability and size-invariance of prepatterns in developing systems. The analysis of synchronization in cellular networks should provide insight into the origin of certain types of cardiac arrhythmias, and may elucidate the role of various types of cell-cell interactions in epileptogenesis.